A Urysohn-like collectionwise separation property on \Psi
by
Jerry E. Vaughan
University of North Carolina at Greensboro

Let Y (also called NÈR by Mrówka) denote the familiar topological space defined using a maximal infinite almost disjoint family A₀ = R of infinite subsets of the natural numbers N. This space consists of the set NÈA₀ in which each natural number n Î N is an isolated point, and each a Î A₀ has a local base consisting of sets of the form {a}È(a\F) where F is a finite subset of N. We will use the notation y(A₀) for this space. Work on the Scarborough-Stone problem leads to the consideration of a larger space denoted y( A₀, A₁) which we call a two step iteration of y. Using this larger space the question under consideration here may be stated simply as follows: Does the Urysohn separation property hold for the space y( A₀, A₁)? This question, however, can be formulated entirely in terms of the space y(A₀) where the Urysohn property of the larger space becomes a certain Urysohn-like collectionwise separation property on y(A₀). Recall that a space satisfies the Urysohn separation property if every pair of points can be separated by closed neighborhoods. A two step iteration of y is a topological space of the form y( A₀, A₁) = y( A₀)È A₁ = NÈA₀ÈA₁, where A₁ is a maximal almost disjoint family of countably infinite subsets of A₀, with the topology in which a local base for a point in NÈA₀ is taken to be the same as in y(A₀), and a local base for a point X Î A₁ consists of all sets of the form

{X}È(X\G)È( È {a\F(a):a Î X\G})

where G is a finite subset of A₀, and F(a) is a finite subset of N for all a Î X\G. Every y( A₀, A₁) is Hausdorff and not regular (provided A₀ is maximal, and A₁ is not empty). We will sketch a proof that assuming h = c, for every A₀ there exists A₁ such that y(A₀, A₁) satisfies the Urysohn separation property. Whether the set theoretic hypothesis "h = c" can be deleted is an open question.

Date received: February 12, 2003